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JEE Main202312 Apr 2023Morning ShiftMathematicsThree Dimensional GeometryActual

Let the lines L 1 : x + 5 3 = y + 4 1 = z - α - 2 and L 2 : 3 x + 2 y + z - 2 = 0 = x - 3 y + 2 z - 13 be coplanar. If the point P a , b , c on L 1 is nearest to the point Q - 4 , - 3 , 2 , then a + b + c is equal to

Options

  1. A12
  2. B14
  3. C8
  4. D10

Correct answer

D. 10

Step-by-step solution

Given, The lines L 1 : x + 5 3 = y + 4 1 = z - α - 2 and L 2 : 3 x + 2 y + z - 2 = 0 = x - 3 y + 2 z - 13 be coplanar, Now direction ratio of line L 2 will be, n → 2 = i ^ j ^ k 3 2 1 1 - 3 2 = i ^ 7 - j ^ 5 + k - 11 Let R lies on L 2 and let z = 0 ∴   3 x + 2 y = 2 and x - 3 y = 13 Now solving above two equation we get, 11 y = – 39 + 2 = – 37 ⇒ y = - 37 11 And x = 13 + 3 y ⇒ x = 13 - 37 × 3 11 ⇒ x = 143 - 111 11 = 32 11 Hence, the point R 32 11 ,   - 37

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